Prevent Google from listing subdomains as part of "primary" domain

I am wanting to ensure that Google (& search engines generally) won't give search results where subdomains are shown as part of the "primary" domain. For example if I google-search primarydomain.com, I don't want to see subdomains or their pages shown as part of primarydomain.com.

I read:
"...Google considers sub domains separate from their parent domains: sub.yoursite.com is considered a different site altogether compared to yoursite.com when it comes to search engine authority."

BUT - I recently googled one of my domains and it turned up with several pages from a subdomain as if they were part of the same website. They are separate and have no links between them.
However, this was in a Parallels set-up where there is no "primary" domain, just an "html_docs" folder in which all domains and subdomains are sub-folders. There is an index.html in the "html_docs" folder which is a placeholder with no links to anything and no domain name pointing to it anyway.

Does anyone know if cPanel can or does prevent search crawlers from treating the "primary" domain folder "public_html" and all its sub-folders containing (in my case) unrelated subdomain websites, as just one big website and jumbling it all together?

I asked the hosting company but they didn't seem to know and said they had "no solution" at this time that would ensure that subdomains were treated independently of the "primary" domain in search results.